Part 1 Find  PK + 1 for the Pk PK=1 + 6 + 11 +...+ [5(k-1)-4] + (5k-4)   Part 2 Prove the inequality for the given integer values of n. n!>2n, n≥4

Q: k! k 1

A: Consider the given series

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Q: The inductive step of an inductive proof shows that for k ≥ 4, if 2 > 3k, then 2+1 ≥ 3(k+1). In…

A: The given proof is .The aim is to find the step for which the fact  is used.

Q: 0.11 Let S(n) = 1 + 2 + + n be the sum of the first n natural numbers and let C(n) = 1³ +2³+...+n³…

A: By the method of mathematical induction we have to show  and .

Q: Proof of Equalities Prove (A BOC) ≤ (ANB) by universal generalization

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Q: How many positive integers n less than or equal to 17,473 satisfy d(n) 1 (mod 2)?

A: We need to find a number of positive integers n≤17473 such that d(n)≡1(mod 2) i.e., , k∈Z.So, we…

Q: ) Prove that if m + n and n+p are even integers, where m, n, and p are integers, then m + p is even.…

A: I am solving first one question as there are many questions asked by student (1) Provided that…

Q: d never list all of the numbers ematical notation, however, w the criterion". mple is the set of all…

A: 7.a) Represent the list of all multiples of 14 We will consider this as n4 where n=1,2,3,4,...Those…

Q: Which inequality is represented in the number line below? 54 3-2-1 0 123 45 OA.-6 <x+2 <-1…

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Q: EXAMPLE Prove that 1 1x2+2x3+ + 3x4 1 4x5 + ... + 1 (n − 1) n n = n-1

A: We have to prove the given statement using mathematical induction.

Q: Let n be any integer. If n=3k+5 for some integer k. Does 91 6n-3 No, because 6n-3= 18k+30-3 which is…

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Q: Consider the following. A = {x | x is a positive integer that is a multiple of 3 and less than 30} B…

A: Given: A=xx is a positive integer that is a multiple of 3 and less than 30B=xx is a positive integer…

Q: 2 For any integer n, n² + 2n + 3 is even if and only if 4|(n² – 2n — 7).

A: What should be done in que 2 should be written clearly. This Que if to be proved seems not true, we…

Q: Prove that for every positive integer n, 72 Σκ2* = (n – 1)2"+1 - k=1

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Q: What is wrong with the following “proof" of the "fact" that n+3 = n+7 for all values of n (besides…

A: Given: Let P(n) be the statement that n+3=n+7 The objective is to prove that P(n) is true for all n…

Q: For all integers a and b, if 5|(a + b), then 5|(a - b).

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Q: For any integer k, 3|(2k² + 1) if and only if 3 | k

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Q: Proposition: If m items are placed into n boxes and n < m, then at least one box contains multiple…

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Q: 2. () Given that a b means Ja+4b , where the positive root i takn,detemine O the value of I 2 (2…

A: We have to solve the operation and solve the equations provided.

Q: Some of the balls are grey and some are white. The balls numbered 2 and 5 are grey. The balls…

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Q: 9 Write an expression for the sum of the first n positive odd integers. 10 Given that k = "(n+1) and…

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Q: (2k+1)! > 10¹

A: